document.write( "Question 1107355: If a quadratic polynomial is divided by (x-2) and (x+1) leave the remainder 11 and 1 respectively, find the polynomial. \n" ); document.write( "

Algebra.Com's Answer #722383 by ikleyn(52915) $\"\"$ $\"About$

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\n" ); document.write( "If a quadratic polynomial is divided by (x-2) and (x+1) leave the remainder 11 and 1 respectively, find the polynomial.
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\n" ); document.write( "\n" ); document.write( "The formulation of this problem is not precisely correct.
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document.write( "    If a quadratic polynomial with the leading coefficient of 1, when divided by (x-2) and (x+1) \r\n" );
document.write( "    leaves the remainder 11 and 1 respectively, find the polynomial.\r\n" );
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document.write( "Let  P(x) =  be this quadratic polynomial with unknown coefficients \"b\" an \"c\".\r\n" );
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document.write( "The fact that  P(x) leaves the remainder 11 when divided by (x-2) is equivalent (according to the \"Remainder Theorem\") that\r\n" );
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document.write( "P(2) = 11,   or    = 11.        (1)\r\n" );
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document.write( "The fact that  P(x) leaves the remainder 1 when divided by (x+1) is equivalent (according to the same theorem) that\r\n" );
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document.write( "P(-1) = 1,   or    = 1.     (2)\r\n" );
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document.write( "Thus we have this system of two equations to find the coefficients \"b\" and \"c\":\r\n" );
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document.write( "2b + c = ,    or   2b + c = 7,     (1')\r\n" );
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document.write( "-b + c = ,   or   -b + c = 0     (2')\r\n" );
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document.write( "To solve the system, subtract eq(2') from eq(1')  (both sides). You will get\r\n" );
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document.write( "2b - (-b) = 7 - 0,   or  3b = 7  ====>  b = .\r\n" );
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document.write( "Then from eq(2')  c = b = .\r\n" );
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document.write( "Thus the quadratic polynomial is  P(x) = .\r\n" );
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document.write( "Check.  P(2) =  =  =  = 4 + 7 = 11,   and\r\n" );
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document.write( "        P(-1) =  =  = 1.   ! Correct !\r\n" );
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document.write( "Answer.  The polynomial is  P(x) = .\r\n" );
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\n" ); document.write( "On Remainder theorem see the lesson\r
\n" ); document.write( "\n" ); document.write( "    - Divisibility of polynomial f(x) by a binomial (x-a) and the Remainder theorem \r
\n" ); document.write( "\n" ); document.write( "in this site.\r
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\n" ); document.write( "\n" ); document.write( "To see other similar solved problems, look into the lesson\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on the Remainder thoerem\r
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\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic
\n" ); document.write( "\"Divisibility of polynomial f(x) by binomial (x-a). The Remainder theorem\".\r
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\n" ); document.write( "\n" ); document.write( "Save the link to this online textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-I
\n" ); document.write( "https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "to your archive and use it when it is needed.\r
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